Data-driven studies of magnetic two-dimensional materials

Trevor David Rhone,1, ∗ Wei Chen,1 Shaan Desai,1 Amir Yacoby,1 and Efthimios Kaxiras1, 2

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA2School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA

(Dated: Friday 22nd June, 2018)

We use a data-driven approach to study the magnetic and thermodynamic properties of vander Waals (vdW) layered materials. We investigate monolayers of the form A2B2X6, based onthe known material Cr2Ge2Te6, using density functional theory (DFT) calculations and machinelearning methods to determine their magnetic properties, such as magnetic order and magneticmoment. We also examine formation energies and use them as a proxy for chemical stability. Weshow that machine learning tools, combined with DFT calculations, can provide a computationallyefficient means to predict properties of such two-dimensional (2D) magnetic materials. Our dataanalytics approach provides insights into the microscopic origins of magnetic ordering in thesesystems. For instance, we find that the X site strongly affects the magnetic coupling betweenneighboring A sites, which drives the magnetic ordering. Our approach opens new ways for rapiddiscovery of chemically stable vdW materials that exhibit magnetic behavior.

I. INTRODUCTION

The discovery of graphene ushered in a new era of stud-ies of materials properties in the two-dimensional (2D)limit [1]. For many years after this discovery only ahandful of van der Waals (vdW) materials were exten-sively studied. Recently, over a thousand new 2D crystalshave been proposed [2, 3]. The explosion in the number ofknown 2D materials increases demands for probing themfor exciting new physics and potential applications [4, 5].Several 2D materials have already been shown to exhibita range of exotic properties including superconductivity,topological insulating behavior and half-metallicity [6–9].Consequently, there is a need to develop tools to quicklyscreen a large number of 2D materials for targeted prop-erties. Traditional approaches, based on sequential quan-tum mechanical calculations or experiments are usuallyslow and costly. Furthermore, a generic approach to de-sign a crystal structure with the desired properties, al-though of practical significance, does not exist yet. Re-search towards building structure-property relationshipsof crystals is in its infancy [10–12].

Long-range ferromagnetism in 2D crystals has recentlybeen discovered [13, 14], sparking a push to understandthe properties of these 2D magnetic materials and todiscover new ones with improved behavior [15–18]. 2Dcrystals provide a unique platform for exploring the mi-croscopic origins of magnetic ordering in reduced dimen-sions. Long-range magnetic order is strongly suppressedin 2D according to the Mermin-Wagner theorem [19], butmagnetocrystalline anisotropy can stabilize magnetic or-dering [20]. This magnetic anisotropy is driven by spin-orbit coupling which depends on the relative positionsof atoms and their identities. As a result, the magneticorder should be strongly affected by changes in the struc-tural arrangements of atoms and chemical composition of

∗ trr715@g.harvard.edu

the crystal.Chemical instability presents a crucial limitation to the

fabrication and use of 2D magnetic materials. For in-stance, black phosphorous degrades upon exposure to airand thus needs to be handled and stored in vacuum or un-der inert atmosphere. Structural stability is a necessaryingredient for industrial scale application of magneticvdW materials, such as CrI3 and Cr2Ge2Te6 [13, 14]. Inaddition to designing 2D materials for desirable magneticproperties, it is important to screen for those materialsthat are chemically stable. In our approach, we employthe calculated formation energy as a proxy for the chem-ical stability [21]. In particular, we obtain the total ener-gies of systems at zero temperature, and calculate the for-mation energy as the difference in total energy betweenthe crystal and its constituent elements in their respec-tive crystal phases. This quantity determines whetherthe structure is thermodynamically stable or would de-compose. This formulation ignores the effects of zero-point vibrational energy and entropy on the stability.

Recently, machine learning (ML) has been combinedwith traditional methods (experiments and ab-initio cal-culations) to advance rapid materials discovery [2, 3, 21–27]. ML models trained on a number of structures canpredict the properties of a much larger set of materi-als. In particular, there is presently a growing inter-est in exploiting ML for discovery of magnetic materi-als [17, 28]. Data-driven studies of ferromagnetism intransition metal alloys have highlighted the importanceof novel data analytics techniques to tackle problems incondensed matter physics [28]. It is conceivable that tun-ing the atomic composition could provide an additionaldegree of freedom in the search for stable 2D materi-als with interesting magnetic properties [29]. Even morecompelling is the ability of ML tools to assist in uncov-ering the physics underlying the stability and magnetismof 2D materials [30, 31]. Specifically, ML methods canidentify patterns in a high-dimensional space revealingrelationships that could be otherwise missed.

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II. METHODOLOGY

In order to develop a path towards discovering 2Dmagnetic materials, we generate a database of structuresbased on a monolayer Cr2Ge2Te6 (Fig. 1(a)) using den-sity functional theory (DFT) calculations [32]. The pos-sible structures amount to a combinatorially large num-ber of type A2B2X6 (∼ 104) with different elements oc-cupying the A, B and X sites. We select a subset of 198structures due to computational constraints. We obtainthe total energy, magnetic order, and magnetic momentof each structure. The ground-state properties were de-termined by examining the energies of the fully optimizedstructure with several spin configurations, including non-spin-polarized, parallel, and anti-parallel spin orienta-tions at the A sites (Fig. 1(b)).

We then employ a set of materials descriptors whichcomprise easily attainable atomic properties, and aresuitable for describing magnetic phenomena. We employadditional descriptors which are related to the formationenergy [33]. The performance of descriptors in predict-ing the magnetic properties or thermodynamic stabilitysheds some light into the origin of these properties.

To create the database we use DFT calculations [34]with the VASP code [35]. We create the different struc-

FIG. 1. (a) Crystal structure of the A2B2X6 lattice. (b)Magnetic orders considered in the A plane, labelled paralleland anti-parallel. (c) Elements used for substitution of A(blue), B (red) and X (magenta) sites.

tures by substituting one of two Cr atoms (A site) in theunit cell with a transition metal atom, from the list: Ti,V, Cr, Mn, Fe, Co, Ni, Cu, Y, Nb, Ru. In the two B siteswe place combinations of Ge, Si, and P atoms, namelyGe2, GeSi, GeP, Si2, SiP, P2. The atoms at X sites wereeither S, Se, or Te, that is, S6, Se6, Te6. Fig 1(c) showsthe choice of substitution atoms in the Periodic Table.An example of a structure created through this processis (CrTi)(SiGe)Te6.

The careful choice of descriptors is essential for the suc-cess of any ML approach [36, 37]. We use atomic proper-ties data from the python mendeleev package 0.4.1 [38] tobuild descriptors for our ML models. We performed su-pervised learning with atomic properties data as inputs,with target properties the magnetic moment and the for-mation energy. The choice of the set of descriptors for themagnetic properties was motivated by the Pauli exclusionprinciple, which gives rise to the exchange and super-

exchange interactions. We also consider the magneto-crystalline anisotropy [39] by building inter-atomic dis-tances and electronic orbital information into our descrip-tors. With respect to the formation energy, the choice ofdescriptors was motivated, in part, by the extended Born-Haber model [33], and include the dipole polarizability,the ionization energy and the atomic radius (see Supple-mental Materials for a full list of atomic properties anddescriptors used [40]).

The data were randomly divided into a training set,a cross-validation set and a test set. Training data andcross-validation were typically 60% of the total data whiletest data comprised 40% of all the data. We employed thefollowing ML models: kernel ridge regression, extra treesregression, and neural networks. Kernel ridge regressionwith a gaussian kernel has been shown to be successfulin several materials informatics studies. Extra trees re-gression allows us to determine the relative importancesof features used in a successful model [41]. An analysisof hidden layers of the deep neural networks could allowus to identify patterns in 2D materials properties data,thereby guiding theoretical studies [31].

III. RESULTS AND DISCUSSION

A. Magnetic properties

We find that the non-spin-polarized configuration hasthe highest energy for all the structures considered. Thatis, all structures prefer either parallel or anti-parallel or-dering in the A plane. Fig. 2(a) shows the energy dif-ference of parallel and anti-parallel spin configurations.Negative (positive) energy difference means the parallel(anti-parallel) is more stable. We note that, because ofthe supercell size limit, we do not consider more com-plex spin configurations in this study. For example, thelowest-energy spin configuration of Cr2Si2Te6 was re-ported to be zigzag anti-ferromagnetic type [42]. Totalmagnetic moments for the lowest energy spin configura-tion of each structure are presented in Fig. 2(b). We findthat only atoms in the A sites show finite magnetic mo-ments, while the moments in the B and X sites are small.Distinct patterns for regions of high and low magneticmoments are observed for X = Te, Se and S in Fig. 2(b).Structures created by substituting non-magnetic atomsat the A site, such as Cu, have small variations in theirrelatively small magnetic moments, as seen in the rowsof Fig. 2(b). However, substitutions of magnetic atoms,such as Mn, result in a set of structures with a large vari-ation in the magnetic moment, with a much larger upperlimit to the range of values observed.

Both the magnetic order and magnetic moment aresensitive to the occupancy of B and X sites, even thoughthe atoms in these sites have negligible contribution tothe overall magnetic moment. Atoms in the X sitesstrongly mediate the magnetic coupling between neigh-boring A sites [42]. Atoms at the B sites can affect the

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FIG. 2. (a) Energy difference between parallel and anti-parallel spin configurations (Eparallel−Eanti-parallel in eV/unitcell) of A2B2X6 structures. (b) Magnetic moment per unitcell (in µB) for each A2B2X6 structure at the lowest energyspin configuration. The occupation of the two B sites is shownon the horizontal axis while that of one of the A site is shownon the vertical axis.

relative positions of A and X sites. Direct exchange be-tween first nearest neighbor A sites competes with super-exchange interactions mediated by the p-orbitals at theX sites. The ground state magnetic order is determinedby the interplay between first, second and third nearestneighbor interactions. Changing the identity of one ofthe A, B or X sites affects the interplay between the di-rect exchange and super-exchange interactions. Recentwork has shown that applying strain to the Cr2Si2Te6lattice tunes the first nearest neighbor interaction, re-sulting in a change in the magnetic ground state fromzig-zag antiferromagnetic to ferromagnetic [42]. Ourwork demonstrates that tuning the composition of theA2B2X6 lattice can have an equivalent effect. For in-

stance, whereas X=Te structures show more parallel ( ¯P )

than anti-parallel (anti- ¯P ) spin-configurations with lowerenergy, there is a clear change when X = Se or S. As Xmoves up the periodic table, there are increasingly moreregions of anti-parallel spin configuration, as well as re-

gions in which ¯P and anti- ¯P are degenerate. In particu-

lar, we find that the distance between nearest neighborA and X sites, as well as two adjacent X sites is linkedto the magnitude of the magnetic moment (see Supple-mental Materials for details).

We use extra trees regression [41] to approximate therelationship between the total magnetic moment and aset of descriptors designed for magnetic property pre-diction (see Supplemental Materials). Training and testdata are considered for the X = Te, Se, and S structuresindividually. The model performance for X = Te is shownin Fig. 3(a). We find reasonable prediction performancefor X = Te that deteriorates for X = Se and is even worsefor X = S. This suggests that our model, along with theset of descriptors used to predict X = Te structures, doesnot generalize well. This could arise due to the fact that

there are more structures that have degenerate ¯P and

anti- ¯P spin configurations if X=Se and S than for X =Te. Nevertheless, subgroup discovery can be exploitedto learn more about these systems [43], implying thatthe identity of the X site strongly affects the magneticproperties of the structures.

FIG. 3. ML predictions of magnetic moments of A2B2X6

structures. (a) Extra trees model performance for the mag-netic moment (in µB) prediction. A subset of structures forX = Te are displayed. The red squares indicate the test data,the green circles show the training data. (b) Top six descrip-tors for the extra trees prediction of the magnetic moment.The size of the bar indicates relative descriptor importance(see text for details).

Determining which descriptors are most important formaking good predictions of a property can be exploitedfor knowledge discovery, especially when a large num-ber of descriptors are available but their relationshipswith the target property are not known [44]. Fig. 3(b)shows the descriptor importances [45] as derived fromextra trees regression. It shows that the ‘the number ofvalence electrons’ [“nvalence max dif” in Fig. 3(b)], ‘theaverage covalent radius’ [“covalentrad avg” in Fig. 3(b)]and the ‘average number of spin up electrons’ [“Nup avg”in Fig. 3(b)], linked to the atomic dipole magnetic mo-ment, are among the top six descriptors in the set exam-ined. The magnetic moment per unit cell is a functionof the magnetic moments of the individual atoms in theunit cell. We examine the local magnetic moments at the

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A sites to determine how the magnetic moment per unitcell is constructed. The local magnetic moment at theA sites (ACr and ATM) can be different from the atomicdipole magnetic moment of the corresponding element.For instance, while the atomic magnetic moment of Cr3+

is 3 µB , the local magnetic moment at ACr fluctuatesfrom 2.7 to 3.2 µB . Fig. 4 (a) shows the local magneticmoment at ATM.

FIG. 4. (a) Local magnetic moment of the transition metalA site, ATM (in µB). (b) Formation energy (in eV/cell) forA2B2X6 structures at the lowest energy spin configuration.Conventions are the same as in Fig. 2.

B. Formation energy

In addition to identifying structures with specific mag-netic properties, the ability to screen for chemical stabil-ity is also important. DFT-calculated formation energies(for the lowest energy spin configuration) are shown inFig. 4 (b). Structures comprising certain elements, suchas Y, decrease the formation energy considerably in com-parison to those without it. Certain transition metals,such as Cu, tend to destabilize the (CrA)B2X6 struc-tures. The formation energy becomes more negative asthe substituted atom at the A site goes from the left tothe right of the first and second row of transition metal el-

ements in the Periodic Table. This is linked to the fillingof the d -orbital, where elements with a filled d -orbital donot form chemical bonds with other elements. Varyingthe composition at the B site does not appear to have astrong impact on the formation energy (see Supplemen-tal Materials, Fig. S1). Changing the X site from Te toSe and then S results in the overall trend of decreasingformation energy.

To exploit the trends in the formation energy data, weuse statistical models to predict the formation energy andto infer structure-property relationships. We find thatsome descriptors, such as the atomic dipole polarizability,are strongly correlated with the formation energy, andare therefore important in generating good ML predic-tions. Since useful descriptors are not always revealed inan analysis of the Pearson correlation coefficient [44], weconsider other methods to learn descriptor importancessuch as the extra trees model [45]. Using the ML mod-

FIG. 5. Formation energy prediction performance of (a) ker-nel ridge regression, (b) deep neural network regression and(c) extra trees regression. Red squares are test data and greencircles training data. (d) Performance of the extra trees re-gression model on the test data as the training set size in-creases, in terms of the R2 and mean absolute error (MAE)scores.

els to predict the formation energy of A2B2X6 structurespermits the quick calculation of the formation energy fora large set of compounds. Whereas DFT calculationsof 104 structures could take up to 1 million CPU hours,the ML prediction takes a few seconds. Fig. 5(a) showsthe prediction performance for kernel ridge regression us-ing a gaussian kernel. Fig. 5(b) shows the performanceof a neural network [46] while Fig. 5(c) shows the per-formance of the extra forests regression. Both trainingset and test set results are displayed, as well as the test

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scores for kernel ridge regression, extra trees regression,and neural network regression.

Further analysis (see Supplemental Materials) showsthat the ‘variance in the ionization energy of atoms’and the ‘average number of valence electrons’ are thetwo most important descriptors in the set examined.This demonstrates a link between the formation energyand the atomic ionization energy, emanating from theincreased atomic ionizability which produces strongerchemical bonding. In addition, the number of valenceelectrons is linked to the number of electrons availablefor bonding. For instance, substitutions by atoms witha filled outer orbital shell will create less stable bonds,leading to chemical instability. The ability of our modelsto generalize is demonstrated by the high scores on thetest data. We further examined how the test set perfor-mance varies with the training set size. Fig. 5(d) showstest scores as a function of training set size using ex-tra trees regression. The test score reaches a plateau atabout a training set size of 40%, with test score (R2) ashigh as 0.91.

C. High-throughput screening using ML models

We can use our trained ML models to make predictionson a wide range of structures not included in the originalDFT data set. Thus far, we have used our ML modelsto estimate the formation energy for an additional 4,223A2B2X6 structures, constructed as follows: (i) For A sitesubstitutions, we considered transition metals not usedin the DFT dataset. (ii) We included Al, Sn and Pb inthe set of atomic substitutions for B sites (not shown).(iii) For the X sites, we added O to our previous choice ofS, Se and Te. The resulting predictions, partly shown inFig. 6(a), provide a means to quickly screen a large dataset of structures for chemical stability. For instance, ourML predictions suggest that structures based on Er, Ta,Hf, Mo, Zr, and Sc in the A site and Al in the B site arelikely to be stable and thus good candidates for furtherexploration.

Magnetic moment predictions are shown in Fig. 6(b).From the results of the ML predictions we select struc-tures with formation energies below -1 eV and mag-netic moments above 5 µB (for X=Te only). From the4,223 predictions, we obtained 40 that satisfied our con-straints. 15 of these were randomly selected for veri-fication with DFT. 5 of these 15 structures were con-firmed to have the expected properties within uncer-tainty. These 15 structures were then added to thetraining data to build an improved model for predict-ing magnetic moment. A second iteration of predic-tion and verification by DFT generated three structures,all of which satisfied the constraints within uncertainty:(CrTc)(SiSn)Te6, (CrTc)Sn2Te6, Cr2(SiP)Te6.

IV. CONCLUSION

We presented evidence that the magnetic properties ofA2B2X6 monolayer structures can be tuned by makingatomic substitutions at A, B, and X sites. This providesa novel framework for investigating the microscopic ori-gin of magnetic order of 2D layered materials and couldlead to insights into magnetism in systems of reduceddimension [13, 14]. Our work represents a path towardtailoring magnetic properties of materials for applicationsin spintronics and data storage [47]. We showed that MLmethods are promising tools for predicting the magneticproperties of 2D magnetic materials. In particular, ourdata-driven approach highlights the importance of the Xsite in determining the magnetic order of the structure.Changing the composition of the A2B2X6 structure altersthe inter-atomic distances and the identity of electronicorbitals. This impacts the interplay between first, secondand third nearest neighbor exchange interactions, whichdetermines the magnetic order.

One goal of this work was to find magnetic 2D ma-terials that are also thermodynamically stable. MLmodels were trained to predict chemical stability thatallow the rapid screening of a large number of pos-sible structures. We showed that the chemical sta-bility of A2B2X6 structures based on Cr2Ge2Te6 canbe tuned by making atomic substitutions. Examplesof structures that satisfy both magnetic moment andformation energy requirements include the following:(CrTc)(SiSn)Te6 and (CrTc)Sn2Te6, not included in ouroriginal DFT database. In addition, we found structuresin our set of DFT calculations that also satisfied our re-quirements: Cr2(SiP)Te6, (TiCr)(SiP)Te6, (YCr)Ge2S6

and (NbCr)Si2Te6.This work provides the impetus for further exploration

of structures with other architectures not consideredhere, that is, with more complex atomic substitutionsbeyond 1 in 2 replacement of Cr atoms at the A site. Weestimate a total number of at least 3×104 structures ofthe A2B2X6 type described in Fig. 1. A computation-ally efficient estimation of the magnetic properties andformation energy is required to quickly explore this vastchemical space. We also expect the ML methods exploredhere, with proper modification, to allow an efficient ex-ploration of other families of 2D magnets, such as CrI3,CrOCl and Fe3GeTe2 [13, 18, 48].

ACKNOWLEDGMENTS

We thank Marios Mattheakis, Daniel Larson, RobertHoyt, Matthew Montemore, Sadas Shankar, Ekin DogusCubuk, Pavlos Protopapas and Vinothan Manoharan forhelpful discussions. For the calculations we used the Ex-treme Science and Engineering Discovery Environment(XSEDE), which is supported by National Science Foun-dation (grant number ACI-1548562) and the Odysseycluster supported by the FAS Division of Science, Re-

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FIG. 6. (a) ML predicted formation energies (in eV/cell) for a wide range of substitutions that were not included in the DFTdata set covering 4,223 new structures (570 are shown here). (b) ML predicted magnetic moments (in µB) for a wide rangeof substitutions that were not included in the DFT data set covering 4,223 new structures (190 are shown here for X=Te).Conventions same as in Fig. 2.

search Computing Group at Harvard University. T.D.R.is supported by the Harvard Future Faculty Leaders Post-

doctoral Fellowship. We acknowledge support from AROMURI Award W911NF-14-0247.

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